Question

Transformations and Similarity Unit Test
2 of 152 of 15 Items



Question
Use the image to answer the question.

A coordinate plane with four quadrants shows an x axis capped with arrows ranging from negative 3 to 12 in 1 unit increments and a y axis capped with arrows ranging from 15 to negative 4 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted in quadrant 1. Point upper A is plotted as left parenthesis 9 comma 9 right parenthesis and point upper B is plotted as left parenthesis 6 comma 6 right parenthesis. The points are plotted on a line segment. Point upper A prime is plotted as left parenthesis 3 comma 3 right parenthesis and upper B prime is plotted as left parenthesis 2 comma 2 right parenthesis. The points are plotted on a line segment.

Determine the scale factor used in the dilation of line segment AB
, with a center point of dilation at the origin.

(1 point)
Responses

3
3

−3
negative 3

−13
negative Start Fraction 1 over 3 End Fraction

13

Answer 1

To determine the scale factor used in the dilation of line segment \( AB \) with a center point of dilation at the origin, we can use the coordinates of points \( A \) and \( B \) and their dilated counterparts \( A' \) and \( B' \).

Given:
- Point \( A(9, 9) \)
- Point \( B(6, 6) \)
- Point \( A'(3, 3) \)
- Point \( B'(2, 2) \)

The scale factor \( k \) is calculated using the formula:
\[
k = \frac{\text{new coordinate}}{\text{original coordinate}}
\]

### For point A:
- Original: \( A(9, 9) \)
- Dilated: \( A'(3, 3) \)

Calculating the scale factor for the x-coordinate:
\[
k = \frac{A'(x)}{A(x)} = \frac{3}{9} = \frac{1}{3}
\]

### For point B:
- Original: \( B(6, 6) \)
- Dilated: \( B'(2, 2) \)

Calculating the scale factor for the x-coordinate:
\[
k = \frac{B'(x)}{B(x)} = \frac{2}{6} = \frac{1}{3}
\]

Both points give us the same scale factor. Thus, the scale factor for the dilation is:
\[
k = \frac{1}{3}
\]

Since the problem states the responses in different formats, note that the scale factor can also be expressed as:
\[
k = -\frac{1}{3} \quad \text{(because dilation can also flip over)}
\]

In this case, if we consider the dilation without the sign, the correct answer based on provided response formats is:
\[
\frac{-1}{3}
\]

So the answer you are looking for is:
**−1/3**